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From: Ben Bacarisse <ben.usenet@bsb.me.uk>
Newsgroups: comp.programming
Subject: Re: are you interested in number theory AND programming
Date: Sat, 13 Jun 2020 17:02:45 +0100
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syedanwarowais5@gmail.com writes:

> As part of a (hopefully growing) exploration in number theory, I need
> to find solutions to the following diophantine equation :
> ab(a+b)=cd(c+d) (in positive integers)
> The first solution set is a=1,b=5 and c=2,d=3.
> I know there are several more solution sets, though I don't know if
> there are an infinite number of solutions sets to this diophantine
> equation.
> Would you like to write a program to solve this puzzle?

It's quite interesting.  Here's a first stab using Haskell (I always use
Haskell for this sort of investigation):

  import System.Environment
   
  psum (a, b) = a * b * (a + b)
   
  get_b_from product a =
    (floor (sqrt(fromIntegral (4*a*product + a^4))) - a^2) `div` (2*a)
   
  possible_pairs_from product =
    takeWhile (\(a, b) -> a <= b) $ map (\a -> (a, get_b_from product a)) [1..]
   
  actual_pairs_from product =
    filter ((==product).psum) $ possible_pairs_from product
   
  solution_list = filter ((>1).length) $ map actual_pairs_from [2,4..]
   
  main = do
    [n] <- getArgs
    mapM (putStrLn.showSolution) $ take (read n) $ solution_list
    where showSolution s = foldl1 (\s1 s2 -> s1 ++ " " ++ s2) $ map show s

The first 10 solution sets this program finds are:

(1,5) (2,3)
(1,14) (3,7)
(1,15) (2,10) (4,6)
(2,13) (3,10)
(1,20) (5,7)
(1,21) (3,11)
(3,15) (6,9)
(2,20) (5,11)
(4,14) (7,9)
(3,17) (5,12)

and there are a few 6-pair solutions that come up:

(1,384) (6,154) (8,132) (14,96) (28,60) (40,44)
(9,231) (12,198) (21,144) (22,140) (42,90) (60,66)
(6,385) (15,238) (23,187) (46,119) (51,110) (69,85)

> Once you respond, I will go deeper into the mathematics this connects
> to.

Probably best not to hold back like this.  Just make it fun with as much
information as you can.  That will encourage more responses.

-- 
Ben.